It is easy to see that to seek a standing wave solution for (REF ), that is, a solution of the form \(\phi (t,x)=u(x)e^{-i\mu t}\) , we need only to get a solution \(u(x)\)  of (REF ) in a suitable function space. When \(s=1\)  and \(N=2\) , the solutions with prescribed \(L^{2}\) -norm (or, saying \(L^2\) -normalized solutions) for (REF ) have attracted much attention in recent years, since in this case the problem is essentially the Gross-Pitaevskii (GP) equation (with mass critical power \(p=\frac{4s}{N}=2\) ), which is a very important model in the study of Bose-Einstein condensate (BEC) theory. For examples, in this case, Bao-Cai [1]} and Guo-Seiringer [2]} studied (REF ) under \(m(x)\equiv 1\)  and \(V(x)\)  satisfies
\(0\le V(x)\in L^{\infty }_{loc}(\mathbb {R}^{2})\ \text{and}\ \lim _{|x|\rightarrow +\infty }V(x)=+\infty ,\) 
